I have never come across any pilot who made it to any major without a degree. I guess you somehow know the lucky few.
In 2007 I was hired by both Northwest and JetBlue and offered an interview at USAirways, all with no degree (albeit with 2 years complete and enrolled in online courses). Ironically, Delta was the only legacy I truly wanted, and I couldn't even apply there without it (and NWA offered the job after I was already doing IOE with JetBlue, so I turned them down. Go figure).
So it's possible, but also limiting.